К 120-летию Дирака

 БОГ ДИРАК

„Религия — это род опиума, который дают народу, чтобы убаюкать его сладкими фантазиями, утешив таким образом насчёт гнетущих его несправедливостей. Недаром всегда так быстро возникает альянс двух важнейших политических сил: государства и церкви. Обе эти силы заинтересованы в сохранении иллюзии, будто добрый боженька если не на земле, то на небе вознаградит тех, кто не возмущался против несправедливостей, а спокойно и терпеливо выполнял свой долг. Вот почему честная констатация того, что этот бог есть просто создание человеческой фантазии, считается худшим смертным грехом.“

Поль Адриен Морис Дира;

Когда возникла задача написать объективную заметку к 120-летию со дня рождения П.А.М. Дирака, проблема эта стала неизбежно усложняться в связи с масштабом личности Дирака. Сейчас обычно читают справку из Википедии: «Дата рождения: 8-ое августа 1902 г.  Дата смерти: 20-ого о;тября 1984 г.  Поль Адриен Морис Дира;; — английс;ий физи;-теорети;, один из создателей ;вантовой теории. Лауреат Нобелевс;ой премии по физи;е 1933 года.

Born: Paul Adrien Maurice Dirac
8 August 1902
Bristol, England
Died: 20 October 1984 (aged 82)
Tallahassee, Florida, U.S.
Nationality: Swiss (1902–19)
British (1919–84)
Alma mater:
University of Bristol
University of Cambridge
Awards:
Nobel Prize in Physics (1933)
Royal Medal (1939)
Copley Medal (1952)
Max Planck Medal (1952)
Fellow of the Royal Society (1930)
Immortal Fellow of The World Platonist’s Academy of Sciences and Arts (Contenant Systems of The World Intellectual Elite Union ) (1984)».

Конечно, внести вклад в квантовую теорию в возрасте 23 лет и удостоиться Нобелевской премии в 31 год при большой скромности, стыдливости и немногословности – признак гениальности. Но если отбросить математические сложности, затрудняющие понимание для читателя (чего стоит хотя бы понимание того, что такое спинор и как им оперировать), и сконцентрироваться просто на перечислении революционных для физики понятий и открытий, подаренных Дираком, удивлению не будет конца. Дирак спустился с небес как Бог, равный Тоту (Гермесу Трисмегисту) и превосходящий полубога Прометея. Что сделал Прометей?
Эсхил в трагедии «Прикованный Прометей» в уста бога Прометея вкладывает фразу: «Смотрите же, что смертным сделал я. Число  им  изобрёл и буквы научил соединять, им память дал, мать муз, всему причину...». Ещё раньше в древнеегипетской мифологии приводится подобное высказывание Бога Тота (Гермеса Трисмегиста). Правда, Бог Амон не счёл дары Тота весьма продуктивными и полезными…На первое место среди своих заслуг перед человечеством Прометей как последователь Гермеса Трисмегиста ставит не огонь, который принёс людям, а число, счёт, письменность,  затем «мать муз» и всему причину - память, которую  олицетворяет Мнемосина - муза памяти.
Позже Пифагор и его ученики открыли много удивительного и  таинственного в мире чисел. Они почти поняли, что числа являются высшей абстрактной бесконечностью, программой, Законом, самим Единым (Сверхбогом), образцом Всего (главной парадигмой). Платон, следуя учениям Пифагора и Парменида, написал блестящий диалог «Парменид», в котором не только развёрнута диалектическая логика единого и многого, но показаны горизонты и трудности человеческого и сверхчеловеческого понимания Единого. Истина предстаёт как процесс самопознания Единого, а Единое – как Логос, Программа Всего. Но она оказывается доступной и открывающейся только отдельным гениям, посвящённым, иерофантам. Параллельно мировые религии пытаются построить свой монотеизм, доступный для профанов, но неизбежно скатывающийся к прежним формам язычества в силу ввода посредников (архангелов, ангелов, серафимов, пророков, святых, организаций и групп типа сект, церквей, атрибутов поклонения типа икон и т.п.). Для посвящённых же главным символом абстрактного абсолюта, Разума и движения к Истине становится Число. Математика является единственным языком Единого. Бог мировых религий явно не дотягивает до Единого. Математика даёт всемогущество, возможность предвидения и точных предсказаний на основании расчётов и познания, основанного не на слепой вере, а на сомнениях, пробах и ошибках, наблюдениях и экспериментах, математических моделях и расчётах. Конечно, присутствуют и вербальные, описательные методы, как, например, у Аристотеля. Но побеждает всё же методология абстрактного мышления, подход Пифагора. Мир и его познание держатся на Числах.
Числами занимались и прославились великие гении: Эратосфен, Диофант, Евклид, Ферма, Декарт, Мерсенн, Гаусс, Лежандр, Эйлер и др.
Позже Коперник, Галилей, Кеплер, Ньютон, Максвелл настолько расширили своими идеями, математическими построениями и открытиями горизонты Вселенной, что планета Земля стала «рядовой», крошечной вокруг «ординарной» звезды – Солнца. Почти всё можно объяснить и даже рассчитать! И казалось, что уже больше нечего добавить к такой реалистичной картине. И действительно, заговорили даже о «конце» физики. Но тут «случилась» квантовая механика. Макс Планк, Луи де Бройль, Нильс Бор, Макс Борн, Эрвин Шрёдингер, Зоммерфельд, Вернер Гейзенберг и несколько отделённые от них Поль-Адриен-Морис Дирак и Энрико Ферми. И всё же фигура Дирака предстаёт особенно выделенно. ДИРАК НЕ ТОЛЬКО ОТКРЫВАЕТ МАТЕМАТИЧЕСКИ НОВУЮ СТРУКТУРУ  ЕДИНОГО В ВИДЕ АНТИМИРА, НО ВВОДИТ В ФИЗИКУ, В РАСЧЁТЫ ПО КВАНТОВОЙ ЭЛЕКТРОДИНАМИКЕ ФИЗИЧЕСКИЙ ВАКУУМ, ЗАПОЛНЕННЫЙ АНТИЧАСТИЦАМИ.
ЕСЛИ РАНЬШЕ БЕСКОНЕЧНОСТЬ И МНИМЫЕ ЧАСТИ КОМПЛЕКСНЫХ ЧИСЕЛ ИМЕЛИ ЛИШЬ МАТЕМАТИЧЕСКИЙ СМЫСЛ, У ДИРАКА ОНИ СТАЛИ ФИЗИЧЕСКОЙ РЕАЛЬНОСТЬЮ. САМО ЕДИНОЕ   
ПРЕДСТАЛО СОВЕРШЕННО ИНЫМ. Представления о вселенной стали менее детерминированными; в микромире детерминизм исчез совсем, уступив место вероятностным величинам. Представьте себе более менее «удачную» случайную флуктуацию «кипящего» квантового вакуума на поверхности «мыльного» пузырька внутри этого бесконечного небытия-вакуума. Это и есть наша вселенная. И законы её, программа её создают пространство-время, иные иллюзии, причём и сами законы, сама программа изменяются. Если Лаплас, построивший стройное здание Небесной механики («Теории Неба»), уже не нуждался в гипотезе бытия Бога (в контексте мировых религий), то Дирак показал, что Единому бессмысленно «играть в Бога» и вмешиваться в такие мелочи, в совершенно ничтожные детали. Представьте себе, что вирусы в вашем организме выдвигают биохимические и генные пожелания к вам для их собственного выживания и благополучия, надеясь, что вы распознаете, обратите внимание и пойдёте им навстречу. Увы, масштабы, задачи, программы совсем другие. Людям не до индивидуализации вирусных запросов. Богам не до «бытовухи» людей. Разве что последние озабочены высшими целями, достойными внимания богов…
Ради справедливости, в связи с откатами профанических слоёв населения к повышенной или чрезмерной религиозности следует заметить, что великие физики, биологи, естествоиспытатели, математики, естественно, не могли верить, не верят и не будут верить в придуманного по социальным причинам Бога мировых религий в антропоморфном, энергетическом или материальном виде, но они не были и «зряшными» атеистами, поскольку признавали Единое, единую гармонию, истину, доказательное познание и объективное знание. Поэтому некоторые субъективные выпады против откровенных высказываний Дирака о религии и боге не имеют под собой логического обоснования. Кстати, его критическое отношение к «опиуму для народа» не воспрепятствовало избранию Дирака в Папскую академию наук…
Собственно, Дирак оказался не мессией, не пророком (как его в шутку называли), а новой проекцией бога Тота – новым Гермесом, удивительно приоткрывшим нам бесконечную природу Единого. Кстати, Дирак любил использовать методы проективной геометрии. Проецирование квантовой флуктуации во вселенскую «оболочку» и обратное проецирование вселенной в вакуум, в небытие являются основаниями моделирования в новой физике.
Как излагает Дирак, «если электрон находится в электромагнитном поле и мы начали с состояния, соответствующего положительной энергии, то он может   перескочить в состояние с отрицательной энергией под влиянием этого поля. Шрёдингер рассмотрел этот вопрос и предложил внести в уравнение, содержащее электромагнитное поле, небольшую по¬правку, благодаря которой можно будет исключить переходы между состояниями с положительными и отрицательными энергиями. Од¬нако такая поправка разрушила бы красоту самого уравнения. Шрёдингер показал, что при этом согласие теоретических значений уровней энергии водорода с экспериментальными данными почти не нарушится, однако релятивистские свойства теории и вся ее красота будут утрачены. Поэтому подобное изменение неприемлемо. Итак, на какое-то время вопрос об отрицательных энергиях стал серьёзнейшей проблемой. Разумеется, с физической точки зрения они недопустимы; экспериментаторы никогда не видят состояний с отрицательными энергиями. И тогда мне пришло в голову, что раз мы не можем исключить эти состояния, мы должны примириться с их существованием и найти для них подходящую физическую ин¬терпретацию…
Разумную физическую интер¬претацию возникших состояний можно получить, используя другое свойство электронов, которое открыл Паули незадолго до этого. Паули показал, что всю таблицу химических элементов можно есте¬ственным образом объяснить, предположив, что в любом квантовом состоянии не может быть более одного электрона. Здесь имеется в виду полное состояние движения электрона с учетом его спина. Указанное предположение известно как принцип запрета Паули. Это одно из основных свойств электронов. Два электрона никогда не могут находиться в одном состоянии.
Предположим теперь, что в известном нам физическом мире все состояния с отрицательными энергиями содержат по одному элект¬рону. Нам придется потребовать бесконечно высокой плотности электронов, потому что состояния с отрицательными энергиями су¬ществуют неограниченно, до минус бесконечности. Электроны, за-полняющие состояния с отрицательными энергиями, можно пред¬ставлять себе как бездонное море. В действительности не так уж трудно вообразить бездонное море. Мы интересуемся только тем, что происходит вблизи поверхности, и можем вполне разумно опи¬сывать такие процессы, забыв о бесконечной плотности электронов, лежащих ниже интересующего нас уровня.
Надо предположить, что по какой-то причине эти электроны не дают вклада в электромагнитное поле.
Мы должны считать, что в вакууме все состояния с положительными энергиями свободны, а состояния с отрицательными энергиями за¬няты, и в плотность электрического заряда дают вклад лишь отклонения от указанного вакуумного распределения…
Переход электрона в состояние с отрицатель¬ной энергией возможен только в том случае, когда в распределении состояний с отрицательными энергиями возникает дырка. При этом и электрон, и дырка исчезают, а их энергия должна проявиться в какой-то иной форме. Эти дырки в распределении электронов с от¬рицательными энергиями являются новым свойством нашей теории, и надо найти их физический смысл…
Нужно усвоить, что вакуум — это область, где все состояния с отрицательными энергиями заполнены. Обычно пола¬гают, что вакуум — это область пространства, где вовсе ничего нет, но более правильно считать, что вакуум — это область пространст¬ва, находящаяся в состоянии с наименьшей возможной энергией. Если мы заполняем состояния с отрицательной энергией, то полная энергия уменьшается. Чем больше таких состояний заполнено, тем ниже должна быть полная энергия, и когда все они заполнены, энергия достигает минимального значения. Таким образом, наша картина вакуума соответствует состоянию, в котором данная об¬ласть пространства обладает наименьшей энергией, что вполне ра¬зумно с физической точки зрения...
Я прекрасно знал, что между массами протона и электрона огромная разница, но думал, что каким-то образом ку¬лоновское взаимодействие между электронами в море может привес¬ти к изменению массы покоя дырки. В результате я опубликовал свою статью на эту тему как теорию электронов и протонов.
Однако очень скоро я столкнулся с оппозицией со стороны мате¬матиков, прежде всего Г. Вейля, который категорически заявил, что новая частица должна иметь ту же массу покоя, что и электрон, и потому она должна соответствовать какому-то объекту, неизвест-ному в физике того времени.
Через несколько лет была открыта новая частица с массой электрона и положительным зарядом. Блэкетт, который работал в то время в Кембридже, первый сказал мне, что у него есть данные о новой частице.
Эти данные были получены при изучении фотографий заряжен¬ных частиц, движущихся в магнитном поле, которое закручивает их траектории, в камере Вильсона. Положительные и отрицатель¬ные частицы закручиваются полем в противоположные стороны. Если просто посмотреть на искривлённый трек на фотографии, то нельзя сказать, оставлен ли он положительно заряженной частицей или же отрицательно заряженной, но движущейся в противополож¬ном направлении. Исследователи нередко замечали, что если интер¬претировать все треки как отрицательно заряженные электроны, то иногда встречаются треки, ведущие в радиоактивный источник. Они считали это простым совпадением. Но Блэкетт, изучавший этот во¬прос, обнаружил, что такие события происходят слишком часто, чтобы быть совпадением, и потому наблюдаемые треки должны ин¬терпретироваться как положительные частицы, испускаемые источ-ником.
Блэкетт не хотел публиковать свои результаты без основатель¬ной проверки, но пока он их проверял, Андерсон совершенно неза¬висимо опубликовал свой опыт, доказывающий, что позитрон дей¬ствительно существует...
Всегда существует возможность родить пару, состоящую из час¬тицы и античастицы. Возмущая вакуумное распределение, мы мо¬жем поднять одну частицу из состояния с отрицательной энергией в состояние с положительной энергией, получив при этом наблю¬даемую частицу с положительной энергией и дырку. Рождается пара — частица и античастица. Возможен также обратный процесс. Если есть античастица, то обычная частица может заполнить дыр¬ку, причем частица и античастица исчезнут, а их энергия перейдет в какую-то другую форму.
В случае электрона и позитрона энергия, требуемая для рожде¬ния пары, составляет примерно миллион вольт. В случае протона и антипротона или нейтрона и антинейтрона эта энергия намного боль¬ше, порядка тысячи миллионов вольт, но существующие в настоя¬щее время ускорители позволяют достичь этой энергии. Уже уда¬лось получить антипротоны и антинейтроны и даже более сложное антивещество…
Появляется также возможность поляризации вакуума. Вакуум¬ное распределение электронов с отрицательными энергиями может быть нарушено под действием электрического или магнитного поля, при этом происходит своеобразная поляризация пространства. И все эти новые идеи возникли из релятивистского волнового уравнения для электрона.
Соединение теории относительности с квантовой механикой дало значение спина, равное половине кванта действия. В настоящее время известно много частиц, спины которых имеют другие значе¬ния. Особый интерес, конечно, представляет фотон, спин которого равен единице. Что нам делать с этими частицами? Это серьезная проблема.
Квантовую теорию для таких частиц можно построить, рассмат¬ривая состояния с какой-либо выделенной осью времени, и выяс¬нить закон изменения этих состояний при повороте этой оси. Здесь, однако, мы встречаемся с той трудностью, что такие изменения нелокальны. Для ансамбля частиц можно построить полевые вели¬чины, изменяющиеся локальным образом, но при переходе к вероят¬ностям для отдельных частиц мы вновь сталкиваемся с нелокальностью.
По-моему, с духом теории относительности не согласуется теория, использующая величины с нелокальным законом преобразования. Это значит, что мы берём некоторую величину в пространстве-вре¬мени с каким-то направлением временной оси, а при повороте этой оси получаем новую величину, относящуюся не только к непосред-ственной окрестности той точки, где была задана исходная величи¬на, но и к областям пространства, отдалённым от неё в той или иной степени. Ситуация отчасти напоминает теорию дальнодействия. Это не очень хорошо с точки зрения теории относительности, но ни¬чего лучшего у нас в настоящее время нет…»
Меня часто просят рассказать о встречах с живым Дираком. Можно рассказывать о встречах с Вернером Гейзенбергом, но о встречах с Дираком – невозможно. Дело в том, что на встречах его фактически не было: он улетал и витал в своём математическом мире. Иногда он читал мысли, причём математические. Как-то он решил посмотреть мои уравнения, нелинейные преобразования и таблицы по вычислению спектра масс элементарных частиц, хотя я их тогда ещё никому не показывал и о них не писал и не говорил. «Я так и думал, - сказал он, - но это поймут только через тысячи лет. Хельмут Хассе написал мне о твоём интересном обобщении великой теоремы Ферма. Но в теории чисел я слаб, хотя и увидел связь с будущей алгеброй многочленов от многих переменных». И, напротив, с Дираком общаешься вживую, когда читаешь его труды и производишь математические операции вслед за ним, в постоянном мысленном диалоге с ним.
Можно было бы ещё много рассказывать о Дираке. Но лучше дать общую справку о его работах и привести его собственные высказывания в оригинале.

Paul Adrien Maurice Dirac was a theoretical physicist who is regarded as one of the most great physicists of the 20th century.
Dirac made fundamental contributions to the early development of both quantum mechanics and quantum electrodynamics. Among other discoveries, he formulated the Dirac equation which describes the behaviour of fermions and predicted the existence of antimatter. Dirac shared the 1933 Nobel Prize in Physics with Erwin Schr;dinger "for the discovery of new productive forms of atomic theory". He also made significant contributions to the reconciliation of general relativity with quantum mechanics.
Dirac was regarded by his friends and colleagues as unusual in character. In a 1926 letter to Paul Ehrenfest, Albert Einstein wrote of Dirac, "I have trouble with Dirac. This balancing on the dizzying path between genius and madness is awful." In another letter concerning the Compton effect he wrote, "I don't understand Dirac at all."
He was the Lucasian Professor of Mathematics at the University of Cambridge, was a member of the Center for Theoretical Studies, University of Miami, and spent the last decade of his life at Florida State University.
Paul Adrien Maurice Dirac was born in Bristol, England, on 8 August 1902. His father, Charles Adrien Ladislas Dirac, was an immigrant from Saint-Maurice, Switzerland, who worked in Bristol as a French teacher. His mother, Florence Hannah Dirac, n;e Holten, the daughter of a ship's captain, was born in Cornwall, England, and worked as a librarian at the Bristol Central Library. Paul had a younger sister, B;atrice Isabelle Marguerite, known as Betty, and an older brother, Reginald Charles F;lix, known as Felix, who died by suicide in March 1925. 
Charles and the children were officially Swiss nationals until they became naturalised on 22 October 1919. Dirac's father was strict and authoritarian, although he disapproved of corporal punishment. Dirac had a strained relationship with his father, so much so that after his father's death, Dirac wrote, "I feel much freer now, and I am my own man." Charles forced his children to speak to him only in French so that they might learn the language. When Dirac found that he could not express what he wanted to say in French, he chose to remain silent.
Paul Dirac was educated first at Bishop Road Primary School and then at the all-boys Merchant Venturers' Technical College (later Cotham School), where his father was a French teacher. The school was an institution attached to the University of Bristol, which shared grounds and staff. It emphasised technical subjects like bricklaying, shoemaking and metal work, and modern languages. This was unusual at a time when secondary education in Britain was still dedicated largely to the classics, and something for which Dirac would later express his gratitude.
Dirac studied electrical engineering on a City of Bristol University Scholarship at the University of Bristol's engineering faculty, which was co-located with the Merchant Venturers' Technical College. Shortly before he completed his degree in 1921, he sat for the entrance examination for St John's College, Cambridge. He passed and was awarded a ;70 scholarship, but this fell short of the amount of money required to live and study at Cambridge. Despite his having graduated with a first class honours Bachelor of Science degree in engineering, the economic climate of the post-war depression was such that he was unable to find work as an engineer. Instead, he took up an offer to study for a Bachelor of Arts degree in mathematics at the University of Bristol free of charge. He was permitted to skip the first year of the course owing to his engineering degree.
In 1923, Dirac graduated, once again with first class honours, and received a ;140 scholarship from the Department of Scientific and Industrial Research. Along with his ;70 scholarship from St John's College, this was enough to live at Cambridge. There, Dirac pursued his interests in the theory of general relativity, an interest he had gained earlier as a student in Bristol, and in the nascent field of quantum physics, under the supervision of Ralph Fowler.  He completed his PhD in June 1926 with the first thesis on quantum mechanics to be submitted anywhere. He then continued his research in Copenhagen and G;ttingen. In the spring of 1929, he was a visiting professor at the University of Wisconsin–Madison.
In 1937, Dirac married Margit Wigner (Eugene Wigner's sister). He adopted Margit's two children, Judith and Gabriel. Paul and Margit Dirac had two children together, both daughters, Mary Elizabeth and Florence Monica.
Dirac was known among his colleagues for his precise and taciturn nature. His colleagues in Cambridge jokingly defined a unit called a "dirac", which was one word per hour. When Niels Bohr complained that he did not know how to finish a sentence in a scientific article he was writing, Dirac replied, "I was taught at school never to start a sentence without knowing the end of it." 
Dirac himself wrote in his diary during his postgraduate years that he concentrated solely on his research, and stopped only on Sunday when he took long strolls alone.
An anecdote recounted in a review of the 2009 biography tells of Werner Heisenberg and Dirac sailing on an ocean liner to a conference in Japan in August 1929. "Both still in their twenties, and unmarried, they made an odd couple. Heisenberg was a ladies' man who constantly flirted and danced, while Dirac—'an Edwardian geek', as biographer Graham Farmelo puts it—suffered agonies if forced into any kind of socializing or small talk. 'Why do you dance?' Dirac asked his companion. 'When there are nice girls, it is a pleasure,' Heisenberg replied. Dirac pondered this notion, then blurted out: 'But, Heisenberg, how do you know beforehand that the girls are nice?'
Margit Dirac told both George Gamow and Anton Capri in the 1960s that her husband had said to a house visitor, "Allow me to present Wigner's sister, who is now my wife."
Another story told of Dirac is that when he first met the young Richard Feynman at a conference, he said after a long silence, "I have an equation. Do you have one too?"
After he presented a lecture at a conference, one colleague raised his hand and said: "I don't understand the equation on the top-right-hand corner of the blackboard". After a long silence, the moderator asked Dirac if he wanted to answer the question, to which Dirac replied: "That was not a question, it was a comment."
Dirac was also noted for his personal modesty. He called the equation for the time evolution of a quantum-mechanical operator, which he was the first to write down, the "Heisenberg equation of motion". Most physicists speak of Fermi–Dirac statistics for half-integer-spin particles and Bose–Einstein statistics for integer-spin particles. While lecturing later in life, Dirac always insisted on calling the former "Fermi statistics". He referred to the latter as "Bose statistics" for reasons, he explained, of "symmetry".
Heisenberg recollected a conversation among young participants at the 1927 Solvay Conference about Einstein and Planck's views on religion between Wolfgang Pauli, Heisenberg and Dirac. Dirac's contribution was a criticism of the political purpose of religion, which Bohr regarded as quite lucid when hearing it from Heisenberg later. 
Heisenberg's view was tolerant. Pauli, raised as a Catholic, had kept silent after some initial remarks, but when finally he was asked for his opinion, said: "Well, our friend Dirac has got a religion and its guiding principle is 'There is no God, and Paul Dirac is His prophet. “ Everybody, including Dirac, burst into laughter.
Later in life, Dirac's views towards the idea of God were less acerbic. As an author of an article appearing in the May 1963 edition of Scientific American, Dirac wrote:
“It seems to be one of the fundamental features of nature that fundamental physical laws are described in terms of a mathematical theory of great beauty and power, needing quite a high standard of mathematics for one to understand it. You may wonder: Why is nature constructed along these lines? One can only answer that our present knowledge seems to show that nature is so constructed. We simply have to accept it. One could perhaps describe the situation by saying that God is a mathematician of a very high order, and He used very advanced mathematics in constructing the universe. Our feeble attempts at mathematics enable us to understand a bit of the universe, and as we proceed to develop higher and higher mathematics we can hope to understand the universe better”.
In 1971, at a conference meeting, Dirac expressed his views on the existence of God. Dirac explained that the existence of God could be justified only if an improbable event were to have taken place in the past:
“It could be that it is extremely difficult to start life. It might be that it is so difficult to start a life that it has happened only once among all the planets... Let us consider, just as a conjecture, that the chance of life starting when we have got suitable physical conditions is 10;100. I don't have any logical reason for proposing this figure, I just want you to consider it as a possibility. Under those conditions ... it is almost certain that life would not have started. And I feel that under those conditions it will be necessary to assume the existence of a god to start off life. I would like, therefore, to set up this connection between the existence of a god and the physical laws: if physical laws are such that to start off life involves an excessively small chance so that it will not be reasonable to suppose that life would have started just by blind chance, then there must be a god, and such a god would probably be showing his influence in the quantum jumps which are taking place later on. On the other hand, if life can start very easily and does not need any divine influence, then I will say that there is no god”.
Dirac did not commit himself to any definite view, but he described the possibilities for scientifically answering the question of God.
Dirac established the most general theory of quantum mechanics and discovered the relativistic equation for the electron, which now bears his name. The remarkable notion of an antiparticle to each fermion particle – e.g. the positron as antiparticle to the electron – stems from his equation. He was the first to develop quantum field theory, which underlies all theoretical work on sub-atomic or "elementary" particles today, work that is fundamental to our understanding of the forces of nature. He proposed and investigated the concept of a magnetic monopole, an object not yet known empirically, as a means of bringing even greater symmetry to James Clerk Maxwell's equations of electromagnetism.
Dirac's first step into a new quantum theory was taken late in September 1925. Ralph Fowler, his research supervisor, had received a proof copy of an exploratory paper by Werner Heisenberg in the framework of the old quantum theory of Bohr and Sommerfeld. Heisenberg leaned heavily on Bohr's correspondence principle but changed the equations so that they involved directly observable quantities, leading to the matrix formulation of quantum mechanics. Fowler sent Heisenberg's paper on to Dirac, who was on vacation in Bristol, asking him to look into this paper carefully.
Dirac's attention was drawn to a mysterious mathematical relationship, at first sight unintelligible, that Heisenberg had established. Several weeks later, back in Cambridge, Dirac suddenly recognized that this mathematical form had the same structure as the Poisson brackets that occur in the classical dynamics of particle motion. At the time, his memory of Poisson brackets was rather vague, but he found E. T. Whittaker's Analytical Dynamics of Particles and Rigid Bodies illuminating. From his new understanding, he developed a quantum theory based on non-commuting dynamical variables. This led him to the most profound and significant general formulation of quantum mechanics to date. Dirac's formulation allowed him to obtain the quantization rules in a novel and more illuminating manner. For this work, published in 1926, Dirac received a PhD from Cambridge. This formed the basis for Fermi-Dirac statistics that applies to systems consisting of many identical spin 1/2 particles (i.e. that obey the Pauli exclusion principle), e.g. electrons in solids and liquids, and importantly to the field of conduction in semi-conductors.
Dirac was famously not bothered by issues of interpretation in quantum theory. In fact, in a paper published in a book in his honour, he wrote: "The interpretation of quantum mechanics has been dealt with by many authors, and I do not want to discuss it here. I want to deal with more fundamental things."
In 1928, building on 2;2 spin matrices which he purported to have discovered independently of Wolfgang Pauli's work on non-relativistic spin systems (Dirac told Abraham Pais, "I believe I got these [matrices] independently of Pauli and possibly Pauli got these independently of me."), he proposed the Dirac equation as a relativistic equation of motion for the wave function of the electron. This work led Dirac to predict the existence of the positron, the electron's antiparticle, which he interpreted in terms of what came to be called the Dirac sea. The positron was observed by Carl Anderson in 1932. Dirac's equation also contributed to explaining the origin of quantum spin as a relativistic phenomenon.
The necessity of fermions (matter) being created and destroyed in Enrico Fermi's 1934 theory of beta decay led to a reinterpretation of Dirac's equation as a "classical" field equation for any point particle of spin ;/2, itself subject to quantization conditions involving anti-commutators. Thus reinterpreted, in 1934 by Werner Heisenberg, as a (quantum) field equation accurately describing all elementary matter particles – today quarks and leptons – this Dirac field equation is as central to theoretical physics as the Maxwell, Yang–Mills and Einstein field equations. Dirac is regarded as the founder of quantum electrodynamics, being the first to use that term. He also introduced the idea of vacuum polarization in the early 1930s. This work was key to the development of quantum mechanics by the next generation of theorists, in particular Schwinger, Feynman, Sin-Itiro Tomonaga and Dyson in their formulation of quantum electrodynamics.
Dirac's The Principles of Quantum Mechanics, published in 1930, is a landmark in the history of science. It quickly became one of the standard textbooks on the subject and is still used today. In that book, Dirac incorporated the previous work of Werner Heisenberg on matrix mechanics and of Erwin Schr;dinger on wave mechanics into a single mathematical formalism that associates measurable quantities to operators acting on the Hilbert space of vectors that describe the state of a physical system. The book also introduced the Dirac delta function. Following his 1939 article, he also included the bra–ket notation in the third edition of his book, thereby contributing to its universal use nowadays.
In 1931, Dirac proposed that the existence of a single magnetic monopole in the universe would suffice to explain the quantization of electrical charge.
Dirac quantized the gravitational field and developed a general theory of the quantum field with dynamical constraints, which forms the basis of the gauge theories and superstring theories of today. The influence and importance of Dirac's work have increased with the decades, and physicists use the concepts and equations that he developed daily.
Dirac's quantum electrodynamics (QED) made predictions that were – more often than not – infinite and therefore unacceptable. A workaround known as renormalization was developed, but Dirac never accepted this. "I must say that I am very dissatisfied with the situation", he said in 1975, "because this so-called 'good theory' does involve neglecting infinities which appear in its equations, neglecting them in an arbitrary way. This is just not sensible mathematics. Sensible mathematics involves neglecting a quantity when it is small – not neglecting it just because it is infinitely great and you do not want it!" His refusal to accept renormalization resulted in his work on the subject moving increasingly out of the mainstream.
However, from his once rejected notes he managed to work on putting quantum electrodynamics on "logical foundations" based on Hamiltonian formalism that he formulated. He found a rather novel way of deriving the anomalous magnetic moment "Schwinger term" and also the Lamb shift, afresh in 1963, using the Heisenberg picture and without using the joining method used by Weisskopf and French, and by the two pioneers of modern QED, Schwinger and Feynman. That was two years before the Tomonaga–Schwinger–Feynman QED was given formal recognition by an award of the Nobel Prize for physics.
Weisskopf and French (FW) were the first to obtain the correct result for the Lamb shift and the anomalous magnetic moment of the electron. At first FW results did not agree with the incorrect but independent results of Feynman and Schwinger. The 1963–1964 lectures Dirac gave on quantum field theory at Yeshiva University were published in 1966 as the Belfer Graduate School of Science, Monograph Series Number, 3. After having relocated to Florida to be near his elder daughter, Mary, Dirac spent his last fourteen years (of both life and physics research) at the University of Miami in Coral Gables, Florida, and Florida State University in Tallahassee, Florida.
In the 1950s in his search for a better QED, Paul Dirac developed the Hamiltonian theory of constraints based on lectures that he delivered at the 1949 International Mathematical Congress in Canada. Dirac had also solved the problem of putting the Schwinger–Tomonaga equation into the Schr;dinger representation  and given explicit expressions for the scalar meson field (spin zero pion or pseudoscalar meson), the vector meson field (spin one rho meson), and the electromagnetic field (spin one massless boson, photon).
The Hamiltonian of constrained systems is one of Dirac's many masterpieces. It is a powerful generalization of Hamiltonian theory that remains valid for curved spacetime.
In the late 1950s, he applied the Hamiltonian methods he had developed to cast Einstein's general relativity in Hamiltonian form  and to bring to a technical completion the quantization problem of gravitation and bring it also closer to the rest of physics according to Salam and DeWitt. In 1959 he also gave an invited talk on "Energy of the Gravitational Field" at the New York Meeting of the American Physical Society. In 1964 he published his Lectures on Quantum Mechanics  which deals with constrained dynamics of nonlinear dynamical systems including quantization of curved spacetime. He also published a paper entitled "Quantization of the Gravitational Field" in the 1967 ICTP/IAEA Trieste Symposium on Contemporary Physics.
Dirac published over 60 papers in last twelve years of his life, including a short book on general relativity. His last paper (1984), entitled "The inadequacies of quantum field theory," contains his final judgment on quantum field theory: "These rules of renormalization give surprisingly, excessively good agreement with experiments. Most physicists say that these working rules are, therefore, correct. I feel that is not an adequate reason. Just because the results happen to be in agreement with observation does not prove that one's theory is correct." The paper ends with the words: "I have spent many years searching for a Hamiltonian to bring into the theory and have not yet found it. I shall continue to work on it as long as I can and other people, I hope, will follow along such lines."


„If we are honest — and scientists have to be — we must admit that religion is a jumble of false assertions, with no basis in reality. The very idea of God is a product of the human imagination.“
Remarks made during the Fifth Solvay International Conference (October 1927), as quoted in Physics and Beyond: Encounters and Conversations (1971) by Werner Heisenberg, pp. 85-86; these comments prompted the famous remark later in the day by Wolfgang Pauli: "Well, our friend Dirac, too, has a religion, and its guiding principle is "God does not exist and Dirac is His prophet."
«If we are honest — and scientists have to be — we must admit that religion is a jumble of false assertions, with no basis in reality. The very idea of God is a product of the human imagination. It is quite understandable why primitive people, who were so much more exposed to the overpowering forces of nature than we are today, should have personified these forces in fear and trembling. But nowadays, when we understand so many natural processes, we have no need for such solutions. I can't for the life of me see how the postulate of an Almighty God helps us in any way. What I do see is that this assumption leads to such unproductive questions as why God allows so much misery and injustice, the exploitation of the poor by the rich and all the other horrors He might have prevented. If religion is still being taught, it is by no means because its ideas still convince us, but simply because some of us want to keep the lower classes quiet. Quiet people are much easier to govern than clamorous and dissatisfied ones. They are also much easier to exploit. Religion is a kind of opium that allows a nation to lull itself into wishful dreams and so forget the injustices that are being perpetrated against the people. Hence the close alliance between those two great political forces, the State and the Church. Both need the illusion that a kindly God rewards — in heaven if not on earth — all those who have not risen up against injustice, who have done their duty quietly and uncomplainingly. That is precisely why the honest assertion that God is a mere product of the human imagination is branded as the worst of all mortal sins».

„One could perhaps describe the situation by saying that God is a mathematician of a very high order, and He used very advanced mathematics in constructing the universe.“
The Evolution of the Physicist's Picture of Nature (1963) .          « It seems to be one of the fundamental features of nature that fundamental physical laws are described in terms of a mathematical theory of great beauty and power, needing quite a high standard of mathematics for one to understand it. You may wonder: Why is nature constructed along these lines? One can only answer that our present knowledge seems to show that nature is so constructed. We simply have to accept it. One could perhaps describe the situation by saying that God is a mathematician of a very high order, and He used very advanced mathematics in constructing the universe. Our feeble attempts at mathematics enable us to understand a bit of the universe, and as we proceed to develop higher and higher mathematics we can hope to understand the universe better».

„I was taught at school never to start a sentence without knowing the end of it.“
http://www-history.mcs.st-and.ac.uk/Printonly/Dirac.html

 „It seems that if one is working from the point of view of getting beauty in one's equations, and if one has really a sound insight, one is on a sure line of progress.“
The Evolution of the Physicist's Picture of Nature (1963): «It seems that if one is working from the point of view of getting beauty in one's equations, and if one has really a sound insight, one is on a sure line of progress. If there is not complete agreement between the results of one's work and experiment, one should not allow oneself to be too discouraged, because the discrepancy may well be due to minor features that are not properly taken into account and that will get cleared up with further development of the theory».

„If you are receptive and humble, mathematics will lead you by the hand.“
As quoted in The Strangest Man: The Hidden Life of Paul Dirac, Mystic of the Atom (2009) by Graham Farmelo, p. 435:  «If you are receptive and humble, mathematics will lead you by the hand. Again and again, when I have been at a loss how to proceed, I have just had to wait until I have felt the mathematics led me by the hand. It has led me along an unexpected path, a path where new vistas open up, a path leading to new territory, where one can set up a base of operations, from which one can survey the surroundings and plan future progress».

„It seems to be one of the fundamental features of nature that fundamental physical laws are described in terms of a mathematical theory of great beauty and power…“
The Evolution of the Physicist's Picture of Nature (1963): «It seems to be one of the fundamental features of nature that fundamental physical laws are described in terms of a mathematical theory of great beauty and power, needing quite a high standard of mathematics for one to understand it. You may wonder: Why is nature constructed along these lines? One can only answer that our present knowledge seems to show that nature is so constructed. We simply have to accept it. One could perhaps describe the situation by saying that God is a mathematician of a very high order, and He used very advanced mathematics in constructing the universe. Our feeble attempts at mathematics enable us to understand a bit of the universe, and as we proceed to develop higher and higher mathematics we can hope to understand the universe better».


„Just by studying mathematics we can hope to make a guess at the kind of mathematics that will come into the physics of the future.“
The Evolution of the Physicist's Picture of Nature (1963): «Just by studying mathematics we can hope to make a guess at the kind of mathematics that will come into the physics of the future. A good many people are working on the mathematical basis of quantum theory, trying to understand the theory better and to make it more powerful and more beautiful. If someone can hit on the right lines along which to make this development, it may lead to a future advance in which people will first discover the equations and then, after examining them, gradually learn how to apply them».

„The interpretation of quantum mechanics has been dealt with by many authors, and I do not want to discuss it here. I want to deal with more fundamental things.“
P. A. M. Dirac, The inadequacies of quantum field theory, in Paul Adrien Maurice Dirac, B. N.
Kursunoglu and E. P. Wigner (Cambridge University, Cambridge, 1987) p. 194.

„The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are… completely known… “
Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character, Vol. 123, No. 792 http://doi.org/10.1098/rspa.1929.0094 (6 April 1929):
           «The underlying physical laws necessary for the mathematical theory of a large part of physics and the whole of chemistry are thus completely known, and the difficulty is only that the exact application of these laws leads to equations much too complicated to be soluble. It therefore becomes desirable that approximate practical methods of applying quantum mechanics should be developed, which can lead to an explanation of the main features of complex atomic systems without too much computation».

„It seems clear that the present quantum mechanics is not in its final form.“
"The Early Years of Relativity" in Albert Einstein : Historical and Cultural Perspectives : The Centennial Symposium in Jerusalem (1979) edited by Gerald James Holton and Yehuda Elkana, p. 85:
            «It seems clear that the present quantum mechanics is not in its final form. Some further changes will be needed, just about as drastic as the changes made in passing from Bohr's orbit theory to quantum mechanics. Some day a new quantum mechanics, a relativistic one, will be discovered, in which we will not have these infinities occurring at all.
It might very well be that the new quantum mechanics will have determinism in the way that Einstein wanted».

„If there is no complete agreement between the results of one's work and the experiment, one should not allow oneself to be too discouraged.“
"The Evolution of the Physicist's Picture of Nature," Scientific American (May, 1963).

„Causality applies only to a system which is left undisturbed. If a system is small, we cannot observe it without producing a serious disturbance and hence we cannot expect to find any causal connexion between the results of our observations.
Causality will still be assumed to apply to undisturbed systems and the equations which will be set up to describe an undisturbed system will be differential equations expressing a causal connexion between conditions at one time and conditions at a later time. These equations will be in close correspondence with the equations of classical mechanics, but they will be connected only indirectly with the results of observations.“
Principles of Quantum Mechanics I. The Principle of Superposition - 1. The Need for a Quantum Theory. - The Principles of Quantum Mechanics (4th ed. 1958).

„The measure of greatness in a scientific idea is the extent to which it stimulates thought and opens up new lines of research.“
The scientific work of Georges Lema;tre (1968), P.A.M. Dirac, Commentarii (Pontifical Academy of Sciences), vol 2, 11, pp. 1–18.

„I want to emphasize the necessity for a sound mathematical basis for any fundamental physical theory. Any philosophical ideas that one may have play only a subordinate role. Unless such ideas have a mathematical basis they will be ineffective.“
The Mathematical Foundations of Quantum Theory (1978).
„My research work was based in pictures. I needed to visualise things and projective geometry was often most useful e. g. in figuring out how a particular quantity transforms under Lorentz transf[ormation]. When I came to publish the results I suppressed the projective geometry as the results could be expressed more concisely in analytic form.“
"Recollections of an Exciting Era," three lectures given at Varenna, 5 August 1972, quoted in Peter Galison, "The Suppressed Drawing: Paul Dirac's Hidden Geometry", Representations, No. 72 (Autumn, 2000).

„At the beginning of time the laws of Nature were probably very different from what they are now. Thus we should consider the laws of Nature as continually changing with the epoch, instead of as holding uniformly throughout space-time. This idea was first put forward by Milne, who worked it out on… assumptions… not very satisfying… we should expect them also to depend on position in space, in order to preserve the beautiful idea of the theory of relativity [that] there is fundamental similarity between space and time.“
The Relation between Mathematics and Physics. http://www.damtp.cam.ac.uk/events/strings02/dirac/speach.html (Feb. 6, 1939) Proceedings of the Royal Society (Edinburgh) Vol. 59, 1938-39, Part II, pp. 122-129.

„Classical mechanics has been developed continuously from the time of Newton and applied to an ever-widening range of dynamical systems, including the electromagnetic field in interaction with matter. The underlying ideas and the laws governing their application form a simple and elegant scheme, which one would be inclined to think could not be seriously modified without having all its attractive features spout.
Nevertheless it has been found possible to set up a new scheme, called quantum mechanics, which is more suitable for the description of phenomena on the atomic scale and which is in some respects more elegant and satisfying than the classical scheme. This possibility is due to the changes which the new scheme involves being of a very profound character and not clashing with the features of the classical theory that make it so attractive, as a result of which all these features can be incorporated in the new scheme.“
Principles of Quantum Mechanics I. The Principle of Superposition - 1. The Need for a Quantum Theory.- The Principles of Quantum Mechanics (4th ed. 1958).

„I don't suppose that applies so much to other physicists; I think it’s a peculiarity of myself that I like to play about with equations, just looking for beautiful mathematical relations which maybe don’t have any physical meaning at all. Sometimes they do.“
              Interview with Dr. P. A. M. Dirac by Thomas S. Kuhn at Dirac's home, Cambridge, England, May 7, 1963 http://www.aip.org/history/ohilist/4575_3.html .

„In science one tries to tell people, in such a way as to be understood by everyone, something that no one ever knew before. But in the case of poetry, it's the exact opposite!“
As quoted in Brighter Than a Thousand Suns : A Personal History of the Atomic Scientists (1958) by Robert Jungk, as translated by James Cleugh, p. 22:
«Anecdotally, when Oppenheimer was working at G;ttingen, Dirac supposedly came to him one day and said: "Oppenheimer, they tell me you are writing poetry. I do not see how a man can work on the frontiers of physics and write poetry at the same time. They are in opposition. In science you want to say something that nobody knew before, in words which everyone can understand. In poetry you are bound to say... something that everybody knows already in words that nobody can understand."

„The aim of science is to make difficult things understandable in a simpler way; the aim of poetry is to state simple things in an incomprehensible way. The two are incompatible.“
As quoted in Dirac: A Scientific Biography (1990), by Helge Kragh, p. 258           [Kragh, Helge, Dirac: A Scientific Biography, https://books.google.co.uk/books...March 30, 1990, 258, December 6, 2017].

„A good deal of my research work in physics has consisted in not setting out to solve some particular problems, but simply examining mathematical quantities of a kind that physicists use and trying to get them together in an interesting way regardless of any application that the work may have. It is simply a search for pretty mathematics. It may turn out later that the work does have an application. Then one has had good luck.“
P.A.M. Dirac, "Pretty Mathematics," International Journal of Theoretical Physics, Vol. 21, Issue 8–9, August 1982, p. 603.  http://link.springer.com/article/10.1007/BF02650229#page-1

„God used beautiful mathematics in creating the world.“
As quoted in The Cosmic Code : Quantum Physics As The Language Of Nature (1982) by Heinz R. Pagels, p. 295; also in Paul Adrien Maurice Dirac : Reminiscences about a Great Physicist (1990) edited by Behram N. Kursunoglu and Eugene Paul Wigner, p. XV .
„One possibility in this direction is to regard, classically, an electron as the end of a single Faraday line of force. The electric field in this picture from discrete Faraday lines of force, which are to be treated as physical things, like strings. One has then to develop a dynamics for such a string like structure, and quantize it…. In such a theory a bare electron would be inconceivable, since one cannot imagine the end of a piece of string without having the string.“
Bombay Lectures (1955).